For a picture of the nation’s border struggles, look at the long lines and understaffed international-passenger checkpoints at Miami International Airport.

Up to 1,000 passengers in a single day have missed connecting flights at the airport — the busiest in the nation for international flights — because they’re held up at the Customs and Border Protection facility.

And the problem could get even worse next month because of looming federal budget cuts, U.S. Department of Homeland Security Secretary Janet Napolitano said Wednesday after visiting the airport.

“Everything we are trying to do here — the additional staffing, overtime, technology…. will come to a screeching halt,” Napolitano said.

If Congress comes up with a deal to avoid the cuts — under the so-called sequester — the agency either needs to shift resources to properly staff its Miami facility or get more money to hire more officers.

Absent a Congressional deal, however, Napolitano also warned that Transportation Security Administration officers might be furloughed as well, meaning travelers should arrive at busy airports like MIA an hour early and citizens re-entering the United States could wind up waiting twice as long to get back in the country.

U.S. Rep. Joe Garcia, a Miami Democrat, toured the airport with Napolitano and said earlier that the situation at the airport encapsulates what’s wrong with Congress as well as the entire immigration system.

“We’re having a big debate over fixing our borders in Mexico, but we can’t even get a rich Argentinian businessman through Customs on time because we don’t have the proper staffing,” Garcia said before the press conference.

Garcia noted that up to half of the estimated 11 million illegal immigrants in the United States overstayed their visas and probably flew into the country — they didn’t cross the U.S.-Mexico border.

“We have to focus on a broader solution,” Garcia said.

While he toured the airport with Napolitano and U.S. Rep. Debbie Wasserman Schultz, Garcia noted that half of the 72 booths at the international-passenger checkpoint were unstaffed on Wednesday.

U.S. Rep. Rep. Ileana Ros-Lehtinen, R-Miami, joined the tour at the airport, but she slipped away before cameras caught her with the Democrats. U.S. Rep. Rep. Mario Diaz-Balart, also a Miami Republican, met Napolitano earlier Wednesday at Port Everglades in Fort Lauderdale but couldn’t make the airport tour because he had engagements in the Naples area. Diaz-Balart called for more officers in South Florida.

After touring the facilities Wednesday, Napolitano pointed out that the lines are better now than in the past.

Almost as soon as the new $180 million facility opened in July, it was understaffed and plagued with long lines.

Gov. Rick Scott wrote to Napolitano in September and again on Wednesday, asking for more Customs and Border Protection staff. He said the long lines and missed flights were bad for Florida’s reputation and therefore its bottom line.

“CBP has not been able to meet the necessary staffing numbers in the new facility. As a result, customers, often numbering well over 1,000 daily, and their baggage are misconnected and must be re-booked on later flights, many leaving the next day,” Scott wrote.

During one 30-day study in February 2012, nearly 5,000 American Airlines customers missed their connecting flights as a direct result of delays in CBP processing.

Napolitano said she was able to provide a few more staff at peak times and that it helped alleviate some of the long lines.

The airport’s director, Jose Abreu, said wait times are down as are the incidence of missed flights. But up to 700 people daily can miss flights because they’re held up in lines, waiting to be processed by border officers.

To further shorten lines, Napolitano said, her agency would like to increase a pre-screening program for low-risk international travelers and hire more border officers in cooperation with local officials.

Miami-Dade Mayor Carlos Gimenez, who met with Napolitano and the congressional members, said he’d be willing to help out by pitching money generated at the airport.

Wasserman Schultz, the Democratic National Committee chairwoman, said she and the other South Florida congressional members will try to pass legislation giving local governments the ability to help underwrite the costs of federal border officers.

But, she said, the looming federal budget cuts need to be handled quickly.

“Deep cuts to Customs and Border Protection operations will mean less staffing and even more frustrated passengers,” she said.